OP if the Royal Free is your closest hospital then you live near me. I would say definitely don't go there. They have a terrible reputation for post natal care, and generally have had a lot of problems with maternity in recent years. A quick search just on this site will bring up posts on it.
I had DS in 2015 and after a lot of research I went to UCH. Yes, it's a bit further away but it is one of the very best hospitals in the country for maternity care. The maternity unit was opened in 2009 so everything is new and light and airy - unlike St Mary's, where my SIL went, which while it has excellent care, the post natal wards were small, boiling hot and not nice places to be.
I ended up having a fairly scary birth as DS got into distress and we were rushed to theatre (I was also consultant lead due to another medical condition) despite the seriousness of what was going on, at no point was I scared; I had absolute faith in the team around us, they were calm, quick, on the ball - and perhaps most significantly, used to dealing with much, much worse. People with complicated pregnancies go from all over the country to UCH because they have specialisms that other hospitals don't have. This made me feel very reassured. They have a NICU, and they are also super close to GOSH if a transfer is needed. I am currently pg with #2 and didn't even consider going anywhere else.
The final nail in the coffin for me was that one of my closest friends works in medical negligence and at one point specialised in maternity. She categorically told me that she would not consider going anywhere else but UCH (which she did, nearly died and they saved her life). That's obviously a bit more anecdotal, and there are other excellent hospitals around (lots of my friends have had good experiences at Queen Charlotte's, and St Mary's is v good just not as nice as UCH post natally)
I don't know about private rooms in UCH post natally, but I was put on a 4 bed ward. I barely saw or heard the others on there, the curtains separated us all the time. I was offered food and water constantly (plus you are on Tott Court Road so most things can be brought to your easily - DH brought me lots of sushi from itsu!) Someone always came quickly if i pressed the buzzer, I was checked lots etc etc. Discharge from hospital was fairly painless. All all it was a very positive experience post birth which I feel grateful for when I hear of what others have gone through. The actual birth wasn't ideal but that was not anything to do with the care i received, which was faultless (DS had the cord round his neck several times)
Hope this is in some way useful, happy for you to pm me if you have any q's 